Resolution of Equations in Algebraic Structures, Volume 1: Algebraic Techniques

462 Pages·1989·23.32 MB·English

by  Hassan Ait-Kaci

Resolution of Equations in
Algebraic Structures
Volume 1
Algebraic Techniques
Edited by
Hassan A'it-Kaci
ACA Systems Technology Laboratory
Programming Languages Group
Microelectronics and Computer Technology Corporation
Austin, Texas
Maurice Nivat
LITP
Universite Paris VII
Paris, France
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
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Copyright © 1989 by Academic Press, Inc.
All rights reserved.
No part of this publication may be reproduced or
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS INC. (LONDON) LTD.
24-28 Oval Road, London NW1 7DX
Library of Congress Cataloging-in-Publication Data
Resolution of equations in algebraic structures / edited by Hassan Ait-
Kaci, Maurice Nivat.
p. cm.
Includes bibliographies and index.
Contents: v. 1. Algebraic techniques - v. 2. Rewriting
techniques.
ISBN 0-12-046370-9 (v. 1). - ISBN 0-12-046371-7 (v. 2)
1. Algebra, Abstract. 2. Equations-Numerical solutions. I. Ait-
Kaci, Hassan, Date- II. Nivat, M.
QA6162.R47 1989
512\02-dc 19 88-21727
CIP
89 90 91 92 9 8 7 6 5 4 3 2 1
Printed in the United States of America
Contents of Volume 2:
Rewriting Techniques
1 Completion Without Failure 1
Leo Bachmair, Nachum Dershowitz, and David A. Plaisted
2 Completion and Its Applications 31
Nachum Dershowitz
3 Extending Equation Solving and Constraint Handling
in Logic Programming 87
M. Dincbas, H. Simonis, and P. Van Hentenryck
4 Proofs by Combinatory Induction
on Recursively Reducible Expressions 117
Laurent Fribourg
5 Completion Algorithms for Conditional Rewriting
Systems 141
Stephane Kaplan and Jean-Luc Remy
6 From Unification in Combination of Equational Theories
to a New AC-Unification Algorithm 171
Claude Kirchner
7 Inductive Completion by Ground Proof Transformation 211
Wolfgang Küchlin
8 Lazy Unification Algorithms for Canonical Rewrite Systems 245
A. Martelli, G. F. Rossi, and C. Moiso
9 Equations in Words 275
Dominique Perrin
10 Order-Sorted Equational Computation 299
Gert Smolka, Werner Nutt, Joseph A. Goguen,
and Jose Mesaguer
VÜ
Contributors
Numbers in parentheses indicate the pages on which the authors' contri­
butions begin.
Egidio Astesiano (1), Dipartimento di Matematica, Universitä de Genova,
via L. B. Alberti 4, 1-16132 Genova, Italy
Shang-Ching Chou (33), Department of Computer Science, University of
Texas, Austin, Texas 78712
Bruno Courcelle (93), Laboratoire d'Informatique Universite Bordeaux-I,
351 cours de la Liberation, F-33405 Talence, France
Desmond Fearnley-Sander (127), Department of Mathematics, University
of Tasmania, GPO Box 252 C, Hobart, Australia 7001
Jean Gallier (151), Department of Computer and Information Science, Uni­
versity of Pennsylvania, Philadelphia, Pennsylvania 19104
Joseph A. Goguen (217) Oxford University Computing Laboratory, Pro­
gramming Research Group, 8-11 Keble Road, Oxford, 0X1 3QD, Eng­
land
Irene Guessarian (263), LITP, UER de Mathematique, Universite de Paris
VII, 2 place Jussieu, F-75251 Paris Cedex 05, France
Paul Mendler (369), Computer Science Department, Manchester Univer­
sity, Manchester Ml 3 9PL, England
Ugo Montanari (293), Istituto di Scienze della Informazione, Universitä di
Pisa, corso Italia 40, 1-56100 Pisa, Italy
Jeffrey F. Naughton (321), Department of Computer Science, Princeton
University, Princeton, New Jersey 08540
Maurice Nivat (351), LITP, Universite de Paris VII, 2 place Jussieu,
F-75251 Cedex 05, France
Prakash Panangaden (369), Department of Computer Science, Cornell Uni­
versity, Upson Hall, Ithaca, New York 14853
ix
χ Contributors
Andreas Podelski (351), LITP, Universite de Paris VII, 2 place Jussieu,
F-75251 Cedex 05, France
Stan Raatz (151), Department of Computer Science, Rutgers University,
New Brunswick, New Jersey 08901
Domenico Sacca (411), viaie deJia Resistenza 198, 1-87306 Renede, Italy
Yehoshua Sagiv (321), Department of Computer Science, Hebrew Univer­
sity of Jerusalem, Givat Ram Campus, Jerusalem, Israel
William F. Schelter (33), Department of Mathematics, University of Texas,
Austin, Texas 78712
Michael I. Schwartzbach (369), Computer Science Department, Aarhus
University, Ny Munkegade, DK-8000 Aarhus C, Denmark
Marcello Sgamma (293), Olivetti DOR, 1-56100 Pisa, Italy
Wayne Snyder (151), Department of Computer and Information Science,
University of Pennsylvania, Philadelphia, Pennsylvania 19104
Martin Wirsing (1), Fakultät für Mathematik und Informatik, Universität
Passau, Postfach 2540, D-8390 Passau, Federal Republic of Germany
Jin-Gen Yang (33), Department of Computer Science, University of Texas,
Austin, Texas 78712
Carlo Zanziolo (411), Microelectronics and Computer Technology Corpora­
tion, ACA Systems Technology Laboratory, 3500 West Balconnes Center
Drive, Austin, Texas 78759
Foreword
Equations are pervasive in computer science. They appear in a wide
variety of algebraic structures, and in a rich diversity of applica­
tions. Therefore, formal methods for establishing the existence of
solutions to equations or for effectively finding such solutions consti­
tute a body of scientific knowledge of fundamental importance. For
as young a discipline as computer science, growing rapidly from sev­
eral shoots into a multitude of branches, the danger of losing sight of
the generality of equational approaches is real. Thus, it is important
to recognize that results obtained in one area of computer science
could be applied in another. The example of fixed-point equations
is blatant, as these manifest themselves virtually everywhere from
programming language semantics and domain theory to deductive
databases. Surveying the state of the art in equation resolution re­
flects more than just a desire to avoid duplication of effort: it is
bound to shed light on those general structures and methods that
may be abstracted from each idiosyncratic application. This is not
only bound to foster cross-fertilization and systematic understanding
of the many equational structures and concepts found to be valuable
in separate branches of computer science, but it is also a vital and
necessary step in the concretization of the discipline that will enable
it to endure as a science.
Having come to this realization, we decided, on the occasion of
a visit of Maurice Nivat to MCC in 1986, to do something about
it. We agreed to organize a colloquium to which we would invite
researchers, working with equations, from diverse areas in computer
science. The idea was to obtain from the best researchers the solid
and stable results of the equational approaches they had devised
or mastered in particular applications. Although the aim was to
bring together experts from several areas of computer science, we
xi
xii Foreword
realized that for the size of the meeting to stay within reason, as
well as for exchanges to be feasible and bear fruit, it would be wise
to limit ourselves to a restricted notion of equations and algebraic
structures—namely, those relevant to symbolic computation and the
foundation of programming. In May of 1987, under the generous
sponsorship of MCC in Austin, Texas, and some partial contribu­
tion from INRIA in Paris, France, the Colloquium on Resolution of
Equations in Algebraic Structures (CREAS) took place in Lakeway,
Texas, with thirty outstanding participants from Australia, France,
Germany, Italy, and the United States. For three overloaded days,
participants exposed and exchanged a wealth of results, ideas, and
prospects.
The meeting was unanimously judged to be a success. There­
fore, it became our duty as organizers to edit a book containing
the highlights of the contributions to CREAS, asking each willing
participant to write a special original piece. Gathering twenty-two
contributions from some of the best researchers working in equation
solving, the book would be meant as a reference, a compendium of
results, methods, algorithms, and state of the art in resolution of
equations in algebraic structures. It would be destined to go on the
shelves of computer science libraries, university departments, and re­
search laboratories, as well as to serve as a fine textbook for graduate
students coming to this field of research.
The resulting collection of papers came to a size too large to fit in
one volume. Therefore, it was decided to divide it into two volumes—
one of which you are holding in your hands. The split was made
easy, as a conceptual line appeared naturally that created a bal­
anced partition between algebraic techniques and, more specifically,
rewriting techniques. Thus, although both volumes share the com­
mon title Resolution of Equations in Algebraic Structures, Volume 1
is subtitled Algebraic Techniques and Volume 2 is subtitled Rewriting
Techniques. More than just a convenient coincidence, the separation
of the specific area of rewriting techniques for equational problems
translates the historical fact that a great deal has happened lately in
this field, which has made it one of the most active fields of research
in symbolic computation. Thus it is only natural to devote a well-
delineated volume to these among all equation-resolution techniques.
Each volume's contents are organized in alphabetical order by first
authors. In each, an introduction reviews each contribution in order
Foreword xiii
of appearance, giving an informal summary of the work and results
reported. The purpose of these reviews is to serve as a quick first-
pass reference to the reader, putting the contribution in context,
stressing the significance of the work.
Finally, we would like to express some thoughts of acknowledge­
ment. We are, of course, indebted to the Advanced Computer Archi­
tecture program of MCC for their generous financial help for organiz­
ing CREAS, as well as for providing their efficient logistic support be­
fore, during, and after the colloquium. In particular, CREAS would
not have been without the encouragement and support of Woody
Bledsoe and Bob Boyer. Many thanks also to INRIA for agreeing to
pay for the travel of their participating researchers. We owe Denise
White (the charming and ever-smiling MCC coordinator of CREAS)
a great deal for taking care of every detail and person without ever
pulling out a shotgun. We were lucky to find at Lakeway Inn in
Lakeway, Texas, two extremely qualified and helpful professionals,
Jo Ann Freeman and Lisa Parker, who provided a most propitious
environment for CREAS. Naturally, we are especially grateful to the
contributors of this volume for gracefully taking the (short) time
to write and polish their articles, and for bending to our editorial
whims. Finally, we acknowledge the kind help and assistance of Sari
Kalin and Alice Peters, of Academic Press Boston. In addition, the
first editor would like to thank Carlo Zaniolo, director of the Lan­
guages Group of MCC's ACA Systems Technology, and the members
of the LIFE project for their understanding and patience during the
interminable time he spent editing these volumes. Most of all, he
needs to express his guilty gratitude to his loving and patient fam­
ily for having foolishly stuck his foot into an unexpected vortex of
never-ending obligations.
We wish you, the reader, a pleasant time sharing the excitement
felt by all who attended CREAS, and hope that we have contributed
in giving you the means and tools to pursue your own research,
solving equations in algebraic structures.
Hassan A'ft-Kaci
Maurice Nivat
A Preview of Volume 1:
Algebraic Techniques
In Bisimulation in Algebraic Specifications, Astesiano and Wirsing
expose an extension of Milner's notion of observational equivalence
of processes in concurrent systems to more abstract algebraic spec­
ifications of transition systems. They build their results upon the
notion of partial abstract types as an appropriate tool for expressing
hierarchies of type enrichments from primitive specifications. Work­
ing from hierarchical specifications consisting of conditional equation
axioms, they construct a complete lattice of simulation congruences
and corresponding fully abstract quotient models. They study in de­
tail the special case of the complete lattice of simulation congruences
associated with the ground atomic theory of hierarchical specifica­
tion, retrieving as the lattice's maximum element Milner's strong
bisimulation for CCS.
In Characteristic Sets and Gröbner Bases in Geometry Theorem
Proving, Chou, Schelter, and Yang report work done in algebraic
methods for geometry theorem proving. Geometric objects are de­
scribed by polynomial equations in cartesian coordinates, translat­
ing the necessary and sufficient constraints that characterize them.
Proving a theorem, also expressed as a polynomial equation, consists
of verifying that it is a consequence of the hypothesis equations.
The authors survey and compare in depth two essential methods,
both based on symbolic polynomial manipulation and both effec­
tively mechanizable. The first one, mainly developed by the Chinese
mathematician Wu Wen-Tsün, uses triangulation and polynomial
pseudo-division. The other, explored by Chou and Schelter among
others, employs a rewrite rule method in conjunction with Buch-
berger's algorithm for finding a Gröbner basis in polynomial ideal
xv
xvi A Preview of Volume 1: Algebraic Techniques
theory. They discuss and illustrate the effectiveness of these meth­
ods on numerous nontrivial theorems of planar Euclidian geometry.
In On Recognizable Sets and Tree Automata, Courcelle exposes the
foundations of an algebraic theory of recognizability as opposed to a
more familiar by less precise and less general notion of algorithmic
recognizability. The former elegantly characterizes a recognizable
subset as a partition of a congruence of finite index, whereas the lat­
ter relies on a cumbersome notion of automaton. Courcelle studies in
detail algebraic recognizability of subsets of free T-algebras, or equa­
tional theories, thus covering the cases of useful discrete structures
like those of words, terms, finite trees, and finite graphs. (Although
not illustrated in this contribution, the fundamental results of this
notion of recognizability are not limited to discrete objects.) Look­
ing at equational presentations as term rewriting systems, syntactic
properties for recognizability in free Γ-algebras are formulated. In
addition, Courcelle extends the familiar determinization nondeter­
ministic (word or tree) automata to special equational theories called
bilinear. These cover all interesting cases of the discrete structures
considered, and allow a connection with an algorithmic notion of
recognizability.
In Idea of a Diagram, Fearnley-Sander proposes a language for
geometric descriptions. He elaborates on the concept of a Grass-
mann Geometry to construct a high-level symbolic language in
which diagrams are represented as sets of polynomial equations.
Grassmann geometry is essentially a first-order language for a
Grassmann Algebra, which is a ring structure over the union
of a field and an affine space. Such a structure allows at once
the manipulation of scalar quantities (elements of the field), points
(elements of the affine space), and vectors (as affine point differ­
ences). Thus, both topological and scalar constraints can be ex­
pressed as polynomial equations. Explicit coordinate manipulation
is conveniently avoided, yielding elegant abstract descriptions. Co­
ordinate-oriented specifications are possible however, reducing Grass-
mann geometry to polynomial algebra. Properties of diagrams in
Grassmann geometry corresponding to elementary geometry theo­
rems are provable using simple equational reasoning. Computation­
ally, the approach is for the most part directly mappable to Horn
logic programming ä la Prolog. A few additional inference rules
like superposition also may be needed. This approach to abstract